In the second month, one pair of rabbits move in, but they don’t have any babies for the first two months. By the sixth month, both the first and second pairs are having a pair of babies every month. The same sequence was named the Fibonacci sequence about 1500 years later.

And just because a series of numbers can be applied to an astonishing variety of objects that doesn’t necessarily imply there’s any correlation between figures and reality. Faces, both human and nonhuman, abound with examples of the Golden Ratio. The mouth and nose are each positioned at golden sections of the distance between the eyes and the bottom of the chin. Similar proportions can been seen from the side, and even the eye and ear itself (which follows along a spiral).

  1. It is possible to find golden ratios in patterns involving circles, triangles, pentagons and other shapes.
  2. From its basis in this structure, the golden rectangle is certainly visually appealing, and it is recognized as one of the most perfect shapes that can be formed.
  3. Drawing a perfect Golden Spiral is pretty tricky to do by hand, but just like with the Golden Ratio, you can get a close approximation with the Fibonacci Spiral.
  4. You might have seen these spirals superimposed over famous pieces of artwork, as experts try and explain why we find them so aesthetically pleasing.

As evidenced by the other names for the number, such as the divine proportion and golden section, many wondrous properties have been attributed to phi. Novelist Dan Brown included a long passage in his bestselling book “The Da Vinci Code” (Doubleday, 2000), in which the main character discusses how phi represents the ideal of beauty and can be found throughout history. Though people have known about phi for a long time, it gained much of its notoriety only in recent centuries. Italian Renaissance mathematician Luca Pacioli wrote a book called “De Divina Proportione” (“The Divine Proportion”) in 1509 that discussed and popularized phi, according to Knott.

The spiral demonstrates that the tiny pointed structures are laid out in a spiral pattern. We can grow this pattern by adding a new, larger square to the long side (a + b) of the rectangle. This square, combined with the previous shapes, results in a new, larger rectangle.

Additionally, if you count the number of petals on a flower, you’ll often find the total to be one of the numbers in the Fibonacci sequence. For example, lilies and irises have three petals, buttercups and wild roses have five, delphiniums have eight petals and so on. Looking at the length of our fingers, each section — from the tip of the base to the wrist — is larger than the preceding one by roughly the ratio of phi.

Sunflower seed head

And when counting these spirals, the total tends to match a Fibonacci number. Interestingly, a highly irrational number is required to optimize filling (namely one that will not be well represented by a fraction). Many falcons, eagles and other raptors follow a golden spiral when attacking their prey — which optimizes their ability to fly and see their prey at the same time as their eyes are at the sides of their heads. That’s the first amazing thing about one of the most famous number sequences in the world — its simplicity. The second fascinating thing about Fibonacci numbers is, like the golden ratio in nature, that we see them everywhere. Interested in the intersection between nature and human architecture?

golden spiral

For instance, phi enthusiasts often mention that certain measurements of the Great Pyramid of Giza, such as the length of its base and/or its height, are in the golden ratio. Others claim that the Greeks used phi in designing the Parthenon or in their beautiful statuary. Mozart made use of the Golden Ratio when writing a number of his piano sonatascloseSonataA piece of instrumental music, usually for a solo instrument, or a small group.. In Mozart’s sonatas, the number of bars of music in the latter section divided by the former is approximately 1.618, the Golden Ratio. Both of the above displayed different algorithms produce geometric constructions that determine two aligned line segments where the ratio of the longer one to the shorter one is the golden ratio. The Fibonacci sequence can also be seen in the way tree branches form or split.

Spiral Leaf Growth

“As the number of primordia increases, the divergence angle eventually converges to a constant value” of 137.5° thereby creating Golden Angle Fibonacci spirals (Seewald). One of the greatest applications of the golden ratio in geometry is the golden rectangle. This quadrilateral figure contains sides that are in proportion to the golden ratio (their ratio and the ratio of the sum of two nonparallel sides to the larger of the parallel sides is equal to 1.618). From its basis in this structure, the golden rectangle is certainly visually appealing, and it is recognized as one of the most perfect shapes that can be formed.

The perfect degree of turn needs to be an irrational number, which can’t be easily approximated by a fraction, and the answer is the Golden Ratio. Le Corbusier explicitly used the golden ratio in his Modulor system for the scale of architectural proportion. Mathematician George Markowsky pointed out that both the Parthenon and the Great Pyramid have parts that don’t conform to the golden ratio, something left out by people determined to prove that Fibonacci numbers exist in everything. Even our bodies exhibit proportions that are consistent with Fibonacci numbers.

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Check out our article on biophilic architecture and interior design or living in a geodesic dome house. The ratio of the length of the longer part, say “a” to the length of the shorter part, say “b” is equal to the ratio of their sum ” (a + b)” to the longer length. We can take the Golden Rectangle one step further by adding a line that forms a quarter circle in each square. But look at the smaller, leftover rectangle shown in pink. This has the same ratio of side lengths as the original rectangle! Even though it’s smaller, it can be divided in the same way as the first.

Since reality is three dimensional, we suggest to look first to see if reality might be written in the code 3D quasicrystals. Time and motion could simply be ordered sequences of different 3D quasicrystal configurations, “played” rapidly like a 3D strip of film frames. Ideas like the invariance of the speed of light could be explained using new ideas like an update of the de Broglie electron clock model.

However, while much of the Parthenon does adhere to the 1.618 ratio very closely, there is no evidence indicating that it was intentional during construction. Despite this, these measurements do make the ancient structure visually alluring, even after the damaging exposure it has succumbed to throughout time. For example, it is intrinsically involved in the internal symmetry of the pentagon, and extends to form part of the coordinates of the vertices of a regular dodecahedron, as well as those of a 5-cell. It features in the Kepler triangle and Penrose tilings too, as well as in various other polytopes. Finally, our favorite example of the is among some of our hardest workers on the planet — bees. Their ancestral tree always follows the Fibonacci sequence of numbers!

The Fibonacci sequence works in nature, too, as a corresponding ratio that reflects various patterns in nature — think the nearly perfect spiral of a nautilus shell and the intimidating swirl of a hurricane. It’s worth noting that every person’s body is different, but that averages across populations tend towards phi. It has also been said that the more closely our proportions adhere to phi, the more “attractive” those traits are perceived. As an example, the most “beautiful” smiles are those in which central incisors are 1.618 wider than the lateral incisors, which are 1.618 wider than canines, and so on. It’s quite possible that, from an evo-psych perspective, that we are primed to like physical forms that adhere to the golden ratio — a potential indicator of reproductive fitness and health.